

A120351


Even numbers k such that the number of odd divisors r and the number of even divisors s are both divisors of k.


2



2, 4, 6, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 34, 36, 38, 44, 46, 48, 52, 58, 62, 68, 72, 74, 76, 80, 82, 86, 90, 92, 94, 106, 112, 116, 118, 120, 122, 124, 126, 134, 142, 144, 146, 148, 150, 158, 160, 164, 166, 168, 172, 176, 178, 180, 188, 192, 194, 198, 202, 206
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OFFSET

1,1


COMMENTS

Since s=0 if k is odd, the number k is necessarily even and then s is always a multiple of r. Note that t=r+s may not be a divisor even if both r and s are divisors. For example, if k=144, then r=3, s=12, but t=r+s=15.


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000


FORMULA

a(n) = n is even, r = number of odd divisors of n, s = number of even divisors of n, are all divisors of n.


EXAMPLE

16 is a term since r=1 and s=4 are both divisors.


MAPLE

with(numtheory); A:=[]: N:=10^4/2: for w to 1 do for k from 2 to N do n:=2*k; S:=divisors(n); r:=nops( select(z>type(z, odd), S) ); s:=nops( select(z>type(z, even), S) ); if andmap(z > n mod z = 0, [r, s]) then A:=[op(A), n]; print(n, r, s); fi; od od; A;


MATHEMATICA

aQ[n_] := Divisible[n, (ev = DivisorSigma[0, n/2])] && Divisible[n, DivisorSigma[0, n]  ev]; Select[Range[2, 206, 2], aQ] (* Amiram Eldar, Nov 02 2019 *)


CROSSREFS

Cf. A033950, A049439, A057265.
Sequence in context: A090127 A057910 A336066 * A022305 A103799 A175817
Adjacent sequences: A120348 A120349 A120350 * A120352 A120353 A120354


KEYWORD

nonn


AUTHOR

Walter Kehowski, Jun 24 2006


EXTENSIONS

Term 2 inserted by Amiram Eldar, Nov 02 2019


STATUS

approved



